sewma.arl | R Documentation |
Computation of the (zero-state) Average Run Length (ARL) for different types of EWMA control charts (based on the sample variance S^2) monitoring normal variance.
sewma.arl(l,cl,cu,sigma,df,s2.on=TRUE,hs=NULL,sided="upper",r=40,qm=30)
l |
smoothing parameter lambda of the EWMA control chart. |
cl |
lower control limit of the EWMA control chart. |
cu |
upper control limit of the EWMA control chart. |
sigma |
true standard deviation. |
df |
actual degrees of freedom, corresponds to subgroup size (for known mean it is equal to the subgroup size, for unknown mean it is equal to subgroup size minus one. |
s2.on |
distinguishes between S^2 and S chart. |
hs |
so-called headstart (enables fast initial response);
the default ( |
sided |
distinguishes between one- and two-sided
two-sided EWMA-S^2 control charts
by choosing |
r |
dimension of the resulting linear equation system (highest order of the collocation polynomials). |
qm |
number of quadrature nodes for calculating the collocation definite integrals. |
sewma.arl
determines the Average Run Length (ARL) by numerically
solving the related ARL integral equation by means of
collocation (Chebyshev polynomials).
Returns a single value which resembles the ARL.
Sven Knoth
S. Knoth (2005), Accurate ARL computation for EWMA-S^2 control charts, Statistics and Computing 15, 341-352.
S. Knoth (2006), Computation of the ARL for CUSUM-S^2 schemes, Computational Statistics & Data Analysis 51, 499-512.
xewma.arl
for zero-state ARL computation of EWMA control charts
for monitoring normal mean.
## Knoth (2005) ## compare with Table 1 (p. 347): 249.9997 ## Monte Carlo with 10^9 replicates: 249.9892 +/- 0.008 l <- .025 df <- 1 cu <- 1 + 1.661865*sqrt(l/(2-l))*sqrt(2/df) sewma.arl(l,0,cu,1,df) ## ARL values for upper and lower EWMA charts with reflecting barriers ## (reflection at in-control level sigma0 = 1) ## examples from Knoth (2006), Tables 4 and 5 Ssewma.arl <- Vectorize("sewma.arl", "sigma") ## upper chart with reflection at sigma0=1 in Table 4 ## original entries are # sigma ARL # 1 100.0 # 1.01 85.3 # 1.02 73.4 # 1.03 63.5 # 1.04 55.4 # 1.05 48.7 # 1.1 27.9 # 1.2 12.9 # 1.3 7.86 # 1.4 5.57 # 1.5 4.30 # 2 2.11 ## Not run: l <- 0.15 df <- 4 cu <- 1 + 2.4831*sqrt(l/(2-l))*sqrt(2/df) sigmas <- c(1 + (0:5)/100, 1 + (1:5)/10, 2) arls <- round(Ssewma.arl(l, 1, cu, sigmas, df, sided="Rupper", r=100), digits=2) data.frame(sigmas, arls) ## End(Not run) ## lower chart with reflection at sigma0=1 in Table 5 ## original entries are # sigma ARL # 1 200.04 # 0.9 38.47 # 0.8 14.63 # 0.7 8.65 # 0.6 6.31 ## Not run: l <- 0.115 df <- 5 cl <- 1 - 2.0613*sqrt(l/(2-l))*sqrt(2/df) sigmas <- c((10:6)/10) arls <- round(Ssewma.arl(l, cl, 1, sigmas, df, sided="Rlower", r=100), digits=2) data.frame(sigmas, arls) ## End(Not run)
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